MASTER AND PUPIL
in series of a differential equation (or system of such equations),
and this solution is only rarely obtainable as a finite expression
in terms of mathematical functions which have been tabulated
(for instance logarithms, trigonometric functions, elliptic func-
tions, etc.). In such problems it then becomes necessary to do
two things: prove that the series converges, if it does; calculate
its numerical values to the required accuracy.
If the series does not converge it is usually a sign that the
problem has been either incorrectly stated or wrongly solved.
The multitude of functions which present themselves in pure
mathematics are treated in the same way, whether they are
ever likely to have scientific applications or not, and finallv a
general theory of convergence has been elaborated to cover vast
tracts of all this, so that the individual examination of a parti-
cular series is often referred to more inclusive investigations
already carried out.
Finally, all this (both pure and applied) is extended to power
series in 2, 3, 4, ... variables instead of the single variable z
above; for example, in two variables,
a -f &os + bjW T- Co32 -f CI^M -f c2zoa + .,. „
It may be said that without the theory of power series most
of mathematical physics (including much of astronomy and
astro-physics) as we know it would not exist.
Difficulties arising with the concepts of limits, continuity,
and convergence drove Weierstrass to the creation of his theory
of irrational numbers.
Suppose we extract the square root of 2 as we did in school,
carrying the computation to a large number of decimal places.
We get as successive approximations to the required square root
the sequence of numbers 1,1.4,1.41,1.412,.....With sufficient
labour, proceeding by well-defined steps according to the usual
rule, we could if necessary exhibit the first thousand, or the
first million, of the rational numbers 1,1.4, ... constituting this
sequence of approximations. Examining this sequence we see
that when we have gone far enough we have determined a
perfectly definite rational number containing as many decimal
places as we please (say 1,000), and that this rational number
differs from any of the succeeding rational numbers in the